Matrix ring over a field and its ideals

Question

Let $M_n(R)$ be the matrix ring over a commutative ring $R$ and let $I$ be an ideal of $R$.

1) Show that if $R=F$ is a field then the only nonzero ideal of $M_n(F)$ is $M_n(F)$ itself.

2) Let $M_n(I)$ be the subset of $M_n(R)$ consisting of matrices with entries in the ideal $I$. Show that $M_n(I)$ is an ideal of $M_n(R)$ and describe the quotient ring $M_n(R)/M_n(I)$ by a matrix ring.

3) Show that if $I$ is a maximal ideal then $M_n(I)$ is a maximal ideal of $M_n(R)$.

Any advice on how to do these questions would be greatly appreciated!

Closely related to http://math.stackexchange.com/questions/362559/show-that-every-ideal-of-the-matrix-ring-m-nr-is-of-the-form-m-ni-where — , May 25 '13 at 16:56

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

1) See Why is the ring of matrices over a field simple? for this.

2) By the way matrix multiplication works, $M_n(R)M_n(I)\subseteq M_n(R)$. One might guess that the quotient is $M_n(R/I)$, so lets look at it backwards: What is the kernel of the canonical homomorphism $M_n(R)\to M_n(R/I)$? (Note that this latter question at the same time shows that $M_n(I)$ is an ideal and what th equotient is)

3) Follows from 1 and 2 as $R/I$ is a field.

edited Apr 13 '17 at 12:20

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answered May 25 '13 at 16:35

Hagen von Eitzen

374,180

@egreg Oh, right.I need to combine row and column operations and linear combine results. Your link gives a much better answer. – Hagen von Eitzen May 26 '13 at 16:19

Matrix ring over a field and its ideals

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